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John Baez (Dec 17 2024 at 23:02):Maybe the problem is that you describe this "obvious construction" in a very detailed way that would require quite a bit of work to follow. Maybe you can compress it down to a sentence or two? Generalizing it might make it easier to understand its key features. When I read
The subobject classifier has two vertices T and F, and five edges: four for each way of including or excluding the source and target while excluding the edge, plus one more for including source, target, and edge.
my eyes glaze over, because I don't know the 'big picture'.
Mike Stay (Dec 17 2024 at 23:33):Thanks for responding. Sorry if it wasn't clear; I thought the Set-enriched examples would be helpful for contextualizing the construction. Here's the short version:
In the arrow topos of functions and commuting squares, the subobject classifier looks like
0 1 2 ΩD domain
| | |
| |/ Ωf
| |
∨ ∨
⊥ ⊤ ΩC codomain
ΩD is a totally ordered set, with 0/"false" < 1/"intermediate" < 2/"true".
Mike Stay (Dec 17 2024 at 23:35):There's a pretty obvious generalization to a Pos-enriched structure (described above) where the truth values ΩD have an entire open unit interval of intermediate values. Does the construction have a name?
Mike Stay (Dec 17 2024 at 23:38):The particular thing you mentioned your eyes glazing over at is this picture you posted on Azimuth:
Mike Stay (Dec 17 2024 at 23:38):
Mike Stay (Dec 17 2024 at 23:42):When we insist that s ≤ t, the top blue edge gets excluded because T (the green vertex, the source of the top blue edge) is not less than or equal to F (the red vertex, the target of the top blue edge), leaving four totally ordered edges.
Mike Stay (Dec 17 2024 at 23:54):We could think of them as the truth values {0, 1/3, 2/3, 1}, though there's no real concept of magnitude beyond the total order.
John Baez (Dec 17 2024 at 23:56):Thanks! I bet some of the topos theorists here could comment.
Mike Shulman (Dec 18 2024 at 05:57):Exactly what construction are you asking whether it has a name? The (1,2)-topos of presheaves on this specific small (1,2)-category? Or its subobject classifier?
Mike Stay (Dec 18 2024 at 06:10):The (1,2)-topos itself, like "the unit interval topos" or "Fuzz" or something. (I presume it has a name; it seems too obvious and simple not to have a common name.)
Mike Shulman (Dec 18 2024 at 06:16):Not a lot of people have studied (1,2)-toposes.
Mike Stay (Dec 18 2024 at 06:16):Huh. OK, thanks.
Mike Shulman (Dec 18 2024 at 06:33):Irrelevantly:
Mike Shulman said:
Exactly what construction are you asking whether it has a name?
This is one of those sentences that I can't figure out how to write it grammatically in English.
The previous sentence should be "This is one of those sentences that I can't figure out how to write grammatically in English" -- the "it" is omitted since "those sentences" were already mentioned earlier. But the question version "Exactly what construction are you asking whether has a name?" sounds wrong to me, although the version with "it" also sounds wrong, and I can't figure out a way to rephrase it that sounds right.
Mike Stay (Dec 18 2024 at 06:54):Ha, I love it! Maybe, "Whether exactly what construction has a name are you asking?"
Or, "About exactly what construction are you asking whether the construction has a name?"
Peva Blanchard (Dec 18 2024 at 06:59):In french, that would be "quelle est cette construction dont tu cherches le nom?"
A literal translation would be "what is that construction you are looking the name for?"
Mike Shulman (Dec 18 2024 at 07:36):Mike Stay said:
Ha, I love it! Maybe, "Whether exactly what construction has a name are you asking?"
Whoo-ee! Makes sense in principle, but while I can imagine pronouncing that aloud so that the listener might have a chance of parsing it correctly (with a rising tone on "what", to make it clear that that's the question word in the question I'm asking), even knowing how it is to be parsed my brain rebels at making sense of it written down. (-:
Or, "About exactly what construction are you asking whether the construction has a name?"
Hmm, that might work. It's a little awkward, but probably clear. Could it be just "About exactly what construction are you asking whether it has a name?"
Nathanael Arkor (Dec 18 2024 at 08:08):It's an interesting grammatical puzzle. Perhaps rephrasing to avoid "it" (without introducing repetition) is clearest?
Regarding which construction are you asking whether there exists a name?
Nathanael Arkor (Dec 18 2024 at 08:09):("About which" might be a better, less formal choice.)
Mike Shulman (Dec 18 2024 at 08:26):Hmm... as a mathematician I feel queasy about a "there exists" that isn't followed by a "such that"... (-:O
Seriously, with that phrasing it doesn't sound as clear to me that the name being asked about is a name for the construction.
Nathanael Arkor (Dec 18 2024 at 08:29):Would "For which" make that clearer, or does it still have the same problem?
Mike Shulman (Dec 18 2024 at 08:34):Ah, that would help! "there exists a name for this" is clearer than "there exists a name regarding this".
Mike Shulman (Dec 18 2024 at 08:35):"For exactly what construction are you asking whether there is a name?"
Kevin Carlson (Dec 18 2024 at 17:30):I think an “it” construction of the form “Of exactly which construction are you asking whether it has a name?” seems to land for me about as well, didn’t see us ruling out that one.
Mike Shulman (Dec 18 2024 at 17:50):Hmm... I wouldn't say "I am asking of that construction whether it has a name" -- that sounds like I went up to the construction and said "hey, do you have a name"?
Mike Shulman (Dec 18 2024 at 17:52):But going back to "about" might work: "about which construction are you asking whether it has a name?" "I am asking about that construction whether it has a name."
Mike Shulman (Dec 18 2024 at 17:54):That's a little more awkward to my ear than Nathanael's suggestion, but I think it works.